Instructor: Hemanshu Kaul
Office: 125C, Rettaliata Engg Center.
E-mail: kaul [at] iit.edu

Class Time: 3:15-4:30pm, Monday and Wednesday
Place: 258, Rettaliata Engg Center

Discussion Forums: Math 332 at Canvas.

Office Hours: Monday and Wednesday at 12:30-1:30pm. And by appointment in-person or through Zoom (send email to setup appointment).
Questions through Canvas Discussion Forums are strongly encouraged.

TA Office Hours: Claude Hall, Monday and Wednesday 10-11am, in RE 129 or through Zoom link at Math Tutoring Center.

ARC Tutoring Service: Mathematics tutoring at the Academic Resource Center.

• Wednesday, 9/4 : Exam#1, Exam#2, and Exam#3 dates have been announced below.


• Monday, 8/19 : Check this webpage regularly for weekly lecture topics, videos, and HW.

• Exam #1 : Wednesday, 9/25. Topics: All the topics corresponding to the HW#1, HW#2, HW#3, HW#4.
• Exam #2 : Wednesday, 10/23. Topics: All the topics corresponding to the HW#5, HW#6, HW#7.
• Exam #3 : Wednesday, 11/20. Topics: All the topics corresponding to the HW#8, HW#9, HW#10.
• Final Exam : Monday, 12/2, 10:30am-12:30, RE 258. Topics: All topics studied during the semester corresponding to HWs #1 through #11.

• Week #1 : 2 Lectures.
  
  • Topics: Linear equations and systems of linear equations, comparison to lines and planes, consistent and inconsistent systems, only three possibilities for number of solutions of a linear system. Matrix notation and terminology, Matrix form of system of linear equations. Augmented matrix. Elementary row operations and back substitution for solving linear systems, Definitions of Row-Echelon and Reduced Row-echelon forms, examples of Row-Echelon and Reduced Row-echelon forms. Gaussian Elimination and Gauss-Jordan algorithms - motivation. correctness and examples. Parametric form of infinite family of solutions, Identifying no solutions, 1 solution and infinitely many solutions from the augmented matrix, Leading 1s, leading and free variables. Homogenous system and its properties - trivial solution and consistency; Homogenous system with more variables then equations has non-trivial solutions (with proof). Algebra of Matrices, Equality of two matrices, Addition, Scalar multiplication. (From Sections 1.1, 1.2, 1.3)
    
  
  
  • Homework: All HW problems below are based on the 12th edition of the textbook. If you are using an earlier edition, please make sure you are solving the correct problems.
    It is essential that you complete the Reading HWs to keep up with the class.
    You only have to submit solutions to `Submission Problems'. However, solving a majority of the `Suggested Problems' is strongly encouraged. Solving these problems will improve your understanding of the course material and better prepare you for the exams.
    Work on the HW problems before and over the weekend so that you ask for help on Canvas forums or in-person with the instructor or the TA during office hours before Wednesday.
    • Reading HW (not for submission):
      Read Example 6 in Section 1.1.
      Read examples for Row-Echelon and Reduced Row-echelon forms in Section 1.2.
      Read Examples 5, 6, 8 in Section 1.2.
      Read Examples 1 to 6 in Section 1.3
      Find 2x2 (and 3x3) matrices A and B such that AB is not equal to BA.
    • Suggested Problems for additional practice (not for submission): Section 1.1: 1, 5, 7, 9, 21&26, TF. Section 1.2: 1, 3, 15, 19, 35, TF. Section 1.3: 23, 30.
    • HW#1 for Submission: Due 11:59pm, Wednesday, 8/28. Submit a PDF file through Canvas Assignment. HW solutions distributed in class.
      You are required to follow the "HW Discussion and Solution Rules" from the Course Information Handout.
      Comment: When solving a linear system, set up the augmented matrix and then apply row operations; clearly label each row operation applied and show all intermediate steps, as we did in class.
      Solve each of the following problems.
      Section 1.1: 12, 16b, 20b, TF(e)(f)(g).
      Section 1.2: 8 and 12 (solve them in continuation/ together), 24ac, 26, 31, 34, 43a.
      Section 1.3: 24 (Just show how to set up the linear system, but then skip the row operations for computing the final values of a, b, c, d).


• Week #2 : 2 Lectures.
  
  • Topics: Homogenous system and its properties - trivial solution and consistency; Homogenous system with more variables then equations has non-trivial solutions (with proof). Algebra of Matrices, Addition and subtraction of matrices, Scalar product of matrices, Product of matrices - condition for definition, relation to dot product, Basic properties of matrix algebra, How to prove Matrix formulas/ identities/ properties. Non-commutativity of Matrix multiplication, Non-properties of Matrix multiplication - Cancelation law and commutativity of product, Zero matrices and their properties, Identity matrices and their properties, Invertible and Singular matrices, Uniqueness of the inverse. (From Sections 1.2, 1.3, 1.4)
    
  
  
  • Homework: All HW problems below are based on the 12th edition of the textbook. If you are using an earlier edition, please make sure you are solving the correct problems.
    It is essential that you complete the Reading HWs to keep up with the class.
    You only have to submit solutions to `Submission Problems'. However, solving a majority of the `Suggested Problems' is strongly encouraged. Solving these problems will improve your understanding of the course material and better prepare you for the exams.
    Work on the HW problems before and over the weekend so that you ask for help on Canvas forums or in-person with the instructor or the TA during office hours before Wednesday.
    • Reading HW (not for submission):
      Read about Partitioned Matrices and Examples 7, 8, 9, and 10 in Section 1.3.
      Read the Matrix form of a Linear system, Definitions 7 and 8, and Examples 11 and 12 in Section 1.3.
      Read Proof of item (d) in Theorem 1.4.1.
      Read Examples 5 and 6 in Section 1.4.
    • Suggested Problems for additional practice (not for submission): Section 1.2: 13, 33. Section 1.3: 1, 3, 5, 7, 8af, 11, 13, 15, 23, 25, 29, 32, TF(e)(f)(l)(m). Section 1.4: 1, 2, 3, 4.
    • HW#2 for Submission: Due 11:59pm, Wednesday, 9/4. Submit a PDF file through Canvas Assignment. HW solutions distributed in class.
      You are required to follow the "HW Discussion and Solution Rules" from the Course Information Handout.
      Solve each of the following problems.
      Section 1.2: 38, 39, 40, TF(b)(d)(g)(i).
      Section 1.3: 5de, 16, 36b, TF(m).
      Section 1.4: 31(b)(c), 54.


• Week #3 : 1 Holiday and 1 Lecture.
  
  • Topics: Invertible and Singular matrices, Uniqueness of the inverse, Inverse of 2X2 matrices, Inverse of product of invertible matrices, Integer powers of a matrix, Laws of exponents for matrices, Properties of transpose, Transpose of AB, Inverse of transpose of an invertible matrix (Reading HW), Elementary matrices - relation with row operations, Method for finding inverse of a matrix and its underlying logic, Finding the inverse of a matrix. (From Sections 1.4, 1.5, and 1.6)
    
  
  
  • Homework: All HW problems below are based on the 12th edition of the textbook. If you are using an earlier edition, please make sure you are solving the correct problems.
    It is essential that you complete the Reading HWs to keep up with the class.
    You only have to submit solutions to `Submission Problems'. However, solving a majority of the `Suggested Problems' is strongly encouraged. Solving these problems will improve your understanding of the course material and better prepare you for the exams.
    Work on the HW problems before and over the weekend so that you ask for help on Canvas forums or in-person with the instructor or the TA during office hours before Wednesday.
    • Reading HW (not for submission):
      Read Example 7 and 8 in Section 1.4.
      Read Example 11 in Section 1.4.
      Read Examples #2, #4 and #5 in Section 1.5.
      
    • Suggested Problems for additional practice (not for submission): Section 1.4: 5, 9, 10, 12, 13, 17, 23, 24, 32, 35, 41, 44, 45, 49. Section 1.5: 1, 3, 5, 7, 9, 13, 19, 20a, 25, 27, 28, 29, 33, TF.
    • HW#3 for Submission: Due 11:59pm, Wednesday, 9/11. Submit a PDF file through Canvas Assignment. HW solutions distributed in class.
      You are required to follow the "HW Discussion and Solution Rules" from the Course Information Handout.
      Solve each of the following problems.
      Section 1.4: 33(a), 36, 40, 43, 50, TF(j)(k).
      Section 1.5: 6b, 8c, 16, 22.


• Week #4 : 2 Lectures.
  
  • Topics: Elementary matrices - relation with row operations, Inverse of Elementary Matrix and their relation to inverse row operations, Statements equivalent to invertibility of a matrix with proofs, Solving linear systems with matrix inversion, Number of solutions of a system of linear equations with proof, Simpler condition for invertibility of a square matrix with proof, for sq matrices AB invertible implies A and B are invertible, Two properties of solutions of non-homogenous systems equivalent to invertibility of a matrix with proofs. Triangular and Diagonal Matrices. Symmetric Matrices. Introduction to Determinants, Properties of determinant under row operations, determinants of triangular matrices and matrices with a zero row or column, det(A)=det(transpose(A)); Using Row operations to evaluate a determinant, Invertibility in terms of determinant, Determinant of product of matrices (with proof), Determinant of the inverse (with proof), Invertibility in terms of determinant with proof (to be completed next week), Determinant of product of matrices with proof (to be completed next week). (From Sections 1.5, 1.6, 1.7, 2.2, 2.3, and elsewhere)
    
  
  
  • Homework: All HW problems below are based on the 12th edition of the textbook. If you are using an earlier edition, please make sure you are solving the correct problems.
    It is essential that you complete the Reading HWs to keep up with the class.
    You only have to submit solutions to `Submission Problems'. However, solving a majority of the `Suggested Problems' is strongly encouraged. Solving these problems will improve your understanding of the course material and better prepare you for the exams.
    Work on the HW problems before and over the weekend so that you ask for help on Canvas forums or in-person with the instructor or the TA during office hours before Wednesday.
    • Reading HW (not for submission):
      Read Examples #3 and #4 in Section 1.6 and pay close attention to the final expression for b.
      Read Theorem 1.6.4 in Section 1.6.
      Do Examples 3-5 from Section 2.1 to practice co-factor expansion for calculating determinant, followed by Example 5 in Section 2.2 for a combination of Row operations and Cofactor expansion.
    • Suggested Problems for additional practice (not for submission): Section 1.6: #5, #15, #18a, #19, #21, T/F#(b)(d). Section 1.7: #1, #9, #13, #17, #21, #25, #30, #37, #47. Section 2.2: #5, #11, #17, #23, #27, #29, #33, TF(d).
    • HW#4 for Submission: Due 11:59pm, Wednesday, 9/18. Submit a PDF file through Canvas Assignment. HW solutions to be distributed in class.
      You are required to follow the "HW Discussion and Solution Rules" from the Course Information Handout.
      Solve each of the following problems.
      Section 1.5: #32.
      Section 1.6: #14, #20, #22, T/F#(f).
      Section 2.2: #20, #24, #26, #34.
      Section 2.3: #34.
      Chapter 2 Supplementary Problems: #33.


• Week #5 and #6 : 3 Lectures and 1 Mid-term Exam.
  
  • Topics: Completion of discussion of - Invertibility in terms of determinant with proof, Determinant of product of matrices. LU decomposition of matrix-- when does it exist and how to find it using row operations in the Gaussian Elimination, Relation between L and elementary matrices and the relation between U and row echelon form, How to use the LU decomposition to easily solve a matrix equation (linear system). Motivation and Definition of vector space, examples and non-examples of Vector Spaces, Examples (R^n, M_{m x n}, F[a,b], P_n, etc.) and non-examples, how to prove V is a vector space, how to prove V is not a vector space - how to show an axiom is not satisfied (Axioms 4 and 5 vs. other axioms). (From Sections 2.2, 2.3, 9.1, 4.1, and elsewhere)
    
  
  
  • Homework: All HW problems below are based on the 12th edition of the textbook. If you are using an earlier edition, please make sure you are solving the correct problems.
    It is essential that you complete the Reading HWs to keep up with the class.
    You only have to submit solutions to `Submission Problems'. However, solving a majority of the `Suggested Problems' is strongly encouraged. Solving these problems will improve your understanding of the course material and better prepare you for the exams.
    Work on the HW problems before and over the weekend so that you ask for help on Campuswire forums or in-person with the instructor or the TA on Monday and Tuesday during office hours.
    • Reading HW (not for submission):
      Do Example 1 and Example 3 in Section 9.1 for an example of direct construction of L and U from the Gaussian elimination procedure.
      [Not part of any Examination Syllabus] Read the Applications of Linear Systems in Section 1.10 based on your interests. In particular, the description of Network Analysis and the Examples 1 and 2 in Section 1.10 for an example of network flows. Also browse through: Use/Purpose of Linear Algebra, by Oliver Knill (Harvard).
      Read Examples 6 and 8 in Section 4.1.
    • Suggested Problems for additional practice (not for submission): Section 2.3: #7, #9, #17, #18, #33, #36, #37, TF(abcdghij). Section 9.1: #1, #5, TF(a)(b). Section 4.1: #1, #2, #5, #6, #9, #10, #12, TF.
    • HW#5 for Submission: Due 11:59pm, Wednesday, 10/2. Submit a PDF file through Canvas Assignment. HW solutions distributed in class.
      You are required to follow the "HW Discussion and Solution Rules" from the Course Information Handout.
      Section 9.1: #6.
      Section 4.1: #4, #7, #8, #16, #19and20, #22, and these two problems.


• Week #7 : 2 Lectures.
  
  • Topics: More examples and non-examples of vector spaces (R^2 with non-standard scalar multiplication, Polynomials of degree=n, Invertible Matrices), how to prove V is a vector space, how to prove V is not a vector space - how to show an axiom is not satisfied (Axioms 4 and 5 vs. other axioms), Some elementary properties of vector spaces with proofs. Introduction to subspaces with examples and non-examples, Characterization of subspaces, Vector space of solution vectors of a homogenous system (Null(A)), Linear combination of vectors, When is vector in R^n a linear combination of some other vectors in R^n? - conversion to a linear system, Span of vectors, Span(S) is a subspace and the smallest subspace containing S, Spanning sets for some vector spaces and subspaces. (From Sections 4.2 and 4.3)
    
  
  
  • Homework: All HW problems below are based on the 12th edition of the textbook. If you are using an earlier edition, please make sure you are solving the correct problems.
    It is essential that you complete the Reading HWs to keep up with the class.
    You only have to submit solutions to `Submission Problems'. However, solving a majority of the `Suggested Problems' is strongly encouraged. Solving these problems will improve your understanding of the course material and better prepare you for the exams.
    Work on the HW problems before and over the weekend so that you ask for help on Canvas forums or in-person with the instructor or the TA during office hours before Wednesday.
    • Reading HW (not for submission):
      Read Example 6, and Examples 7-10 together with the Figure of function spaces in Section 4.2.
      Read Examples 11 and 12 in Section 4.2.
      Read Theorem 4.3.1 in Section 4.3.
      Read Examples 1-6 in Section 4.3.
    • Suggested Problems for additional practice (not for submission): Section 4.2: #1, #2, #3, #4, #5, #6, #9, #10, #11, #12, #15, #16, #24, T/F. Section 4.3: #1, #3, #4, #6, #7, #9, #11, #16.
    • HW#6 for Submission: Due 11:59pm, Wednesday, 10/9. Submit a PDF file through Canvas Assignment. HW solutions distributed in class.
      You are required to follow the "HW Discussion and Solution Rules" from the Course Information Handout.
      Solve each of the following problems.
      Section 4.2: #1c and #2a, #3b and #4a, #7b, #16b, TF(e)(f)(g).
      Section 4.3: #3a, #4a and #6a, #8c, #10, #15a.


• Week #8 : 1 Holiday and 1 Lecture.
  
  • Topics: When is vector in R^n a linear combination of some other vectors in R^n? - conversion to a linear system, Span of vectors, Span(S) is a subspace and the smallest subspace containing S, Spanning sets for some vector spaces and subspaces, Spanning sets for some vector spaces and subspaces, Conversion of a spanning set problem into a linear system problem, linear independence and its motivations, Linear independence and dependence of vectors with examples and non-examples, Relation between a vector equation and a linear system (through examples), Characterization of linear dependence and independence in terms of linear combinations (to be completed). (From Sections 4.3 and 4.4)
    Discussion of EXAM #1 solutions during Special Zoom Office Hour with video recording.
    
  
  
  • Homework: All HW problems below are based on the 12th edition of the textbook. If you are using an earlier edition, please make sure you are solving the correct problems.
    It is essential that you complete the Reading HWs to keep up with the class.
    You only have to submit solutions to `Submission Problems'. However, solving a majority of the `Suggested Problems' is strongly encouraged. Solving these problems will improve your understanding of the course material and better prepare you for the exams.
    Work on the HW problems before and over the weekend so that you ask for help on Canvas forums or in-person with the instructor or the TA during office hours before Wednesday.
    • Reading HW (not for submission):
      Read Examples 6-7 in Section 4.3.
      Read Examples 1-5 and 6 in Section 4.4.
      Read Theorem 4.4.2 in Section 4.4.
    • Suggested Problems for additional practice (not for submission): Section 4.3: #20, #23, TF. Section 4.4: #1, #2, #3, #5, #11, #26, #27, TF.
    • HW#7 for Submission: Due 11:59pm, Wednesday, 10/16. Submit a PDF file through Canvas Assignment. HW solutions distributed in class.
      You are required to follow the "HW Discussion and Solution Rules" from the Course Information Handout.
      Solve each of the following problems.
      Section 4.3: #16a, #19, TF(c)(g).
      Section 4.4: #4a, #6, #10a, #11, #22, one of (#26 or #27), #28.


• Week #9 and #10 : 3 Lectures and 1 Mid-term Exam.
  
  • Topics: Some simple reasons for linear dependence, A sufficient condition for linear dependence in R^n, Basis of a Vector Space, Standard bases for R^n, P_n, and M_nn. How to show S is a Basis of R^n, P_n, etc., Basis of the solution space of a homogenous system, Uniqueness of basis representation, Coordinate vector relative to a basis with examples from R^n and P_n, Properties of sets with more or with less vectors than in a basis, Dimension of a vector space, examples, dimension of the solution space of a homogenous system. Plus/Minus theorem, How to check for basis of a vector space whose dimension is known, Converting a large spanning set or a small linearly independent set into a basis, How to extend a set of vectors into a basis for R^n, dimension of a subspace vs vector space containing it. (From Sections 4.4, 4.5, 4.6)
    
  
  
  • Homework: All HW problems below are based on the 12th edition of the textbook. If you are using an earlier edition, please make sure you are solving the correct problems.
    It is essential that you complete the Reading HWs to keep up with the class.
    You only have to submit solutions to `Submission Problems'. However, solving a majority of the `Suggested Problems' is strongly encouraged. Solving these problems will improve your understanding of the course material and better prepare you for the exams.
    Work on the HW problems before and over the weekend so that you ask for help on Canvas forums or in-person with the instructor or the TA during office hours before Wednesday.
    • Reading HW (not for submission):
      Read Examples 8-9 in Section 4.5.
      Read Theorems 4.6.3, 4.6.4, 4.6.5. 4.6.6, and the Examples 4, 5, in Section 4.6.
      Read and carefully understand the very important statements in Exercises #22 and #23 in Section 4.6. Try proving them.
    • Suggested Problems for additional practice (not for submission): Section 4.5: #3, #7, #9, #11, #13, #14, #15, #17, #26, #28, #29, #TF. Section 4.6: #5, #8, #9, #10, #11, #13, #15, #17, #22 and 23, #TF.
    • HW#8 for Submission: This is a longer HW based on 3 lectures. Note the special submission time. Due 11:59pm, Thursday, 10/31. Submit a PDF file through Canvas Assignment. HW solutions distributed in class.
      You are required to follow the "HW Discussion and Solution Rules" from the Course Information Handout.
      Solve each of the following problems.
      Section 4.5: #4, #6, #10 [HINT: cos^2(x) - sin^2(x) = cos(2x)], #16, #25.
      Section 4.6: #4 [Read example 3 first], #8b, #9a, #10, #11, #14, #18, #TF(c)(i).


• Week #11 : 2 Lectures.
  
  • Topics: Converting a large spanning set or a small linearly independent set into a basis, How to extend a set of vectors into a basis for R^n. Change of basis problem and transition matrix for relating the two coordinate vectors, Relation between the two transition matrices. Row space, Column space, and Null space of a matrix, Relation between consistency of a non-homogenous system and the Column space, General solution of a non-homogenous system in terms of a particular solution and a general solution of the corresponding homogenous system, Row operations and Row, Col and Null spaces of a matrix and their bases, Finding Basis for Row(A), Col(A) and Null(A), Using Row(A) and Col(A) to find a basis of a Euclidean subspace expressed as Span(S) - the difference between the two methods. (From Sections 4.7 and 4.8)
    
  
  
  • Homework: All HW problems below are based on the 12th edition of the textbook. If you are using an earlier edition, please make sure you are solving the correct problems.
    It is essential that you complete the Reading HWs to keep up with the class.
    You only have to submit solutions to `Submission Problems'. However, solving a majority of the `Suggested Problems' is strongly encouraged. Solving these problems will improve your understanding of the course material and better prepare you for the exams.
    Work on the HW problems before and over the weekend so that you ask for help on Canvas forums or in-person with the instructor or the TA during office hours before Wednesday.
    • Reading HW (not for submission):
      Read all Examples in Section 4.7, and make a note of Procedure (14).
      Read Theorems 4.7.1 and 4.7.2 in Section 4.7.
      Read Examples 4, 5, 6, 8 in Section 4.8.
      Read and carefully understand the very important statements in Exercises #20 and #21 in Section 4.7. Try proving them.
    • Suggested Problems for additional practice (not for submission): Section 4.7: #1, #3, #5, #9, #13, #14, #16. Section 4.8: #3, #5, #7a, #11, #14, #33.
    • HW#9 for Submission: Due 11:59pm, Wednesday, 11/6. Submit a PDF file through Canvas Assignment. HW solutions to be distributed in class.
      You are required to follow the "HW Discussion and Solution Rules" from the Course Information Handout.
      Solve each of the following problems.
      Section 4.7: #4, #6, #12.
      Section 4.8: #6, #7b, #10a, #16, #18 [First finish the reading HW (Example 6)], #TF(g)(h), #34 [Hint: TF(g) is true].


• Week #12 : 2 Lectures.
  
  • Topics: Finding Basis for Row(A), Col(A) and Null(A), Using Row(A) and Col(A) to find a basis of a Euclidean subspace expressed as span(S) - the difference between the two methods, Statements with proofs related to: rank(A), nullity(A), Row(A)=Col(A^T), rank(A)=rank(A^T), rank + nullity = #of columns, rank and nullity in terms of the solution of the corresponding homogenous system, Consistency theorem, Equivalent statements for rank(A) = #rows, Consistency properties of linear systems with non-square coefficient matrices. Eigenvalues and eigenvectors of a matrix, Characteristic polynomial and characteristic equation of a matrix; Eigenspace of a matrix w.r.t. an eigenvalue, Finding bases for the eigenspaces of a matrix, Invertibility and eigenvalues. Eigenvector problem and the Diagonalization problem, Definition and motivation for diagonalizability of matrices, Similar matrices, Characterization of diagonalizable matrices in terms of eigenvectors. (From Sections 4.8, 4.9, 5.1 and 5.2)
    Discussion of EXAM #2 solutions during Special Zoom Office Hour with video recording.
    
  
  
  • Homework: All HW problems below are based on the 12th edition of the textbook. If you are using an earlier edition, please make sure you are solving the correct problems.
    It is essential that you complete the Reading HWs to keep up with the class.
    You only have to submit solutions to `Submission Problems'. However, solving a majority of the `Suggested Problems' is strongly encouraged. Solving these problems will improve your understanding of the course material and better prepare you for the exams.
    Work on the HW problems before and over the weekend so that you ask for help on Campuswire forums or in-person with the instructor or the TA on Monday and Tuesday during office hours.
    • Reading HW (not for submission):
      Read Example 4 in Section 4.9.
      Review the discussion under "Bases for the Fundamental Spaces" in Section 4.9 and then read Examples 1 and 5 in the same Section.
      Read Examples 3, 6 and 7 in Section 5.1.
      Read and understand the statements in Table 1 of Section 5.1.
      Read the "Procedure for Diagonalizing an n x n Matrix" followed by Examples 1 and 2 in Section 5.2.
    • Suggested Problems for additional practice (not for submission): Section 4.9: #1, #3, #7, #9, #19, #25, #28 and #29, #39. Section 5.1: #3, #5, #10, #13, #25, #27, #33. Section 5.2: #3, #5, #9, #15, #19, #24, #25, #26, #TF.
    • HW#10 for Submission: Due 11:59pm, Wednesday, 11/13. Submit a PDF file through Canvas Assignment. HW solutions distributed in class.
      You are required to follow the "HW Discussion and Solution Rules" from the Course Information Handout.
      Solve each of the following problems.
      Section 4.9: #6, #7b and#8, #24a, #31, any one of {#40 OR #TF(b)} [Hint: Consider A as m x n matrix and look at linear independence of rows and columns when m less than n and when m larger than n].
      Section 5.1: #8, #24a and#25, #34.
      Section 5.2: #8, #20a, TF(d)(e).


• Weeks #13, #14 and #15 : 4 Lectures, 1 Exam, and 1 holiday.
  
  • Topics: Characterization of diagonalizable matrices in terms of eigenvectors and sum of nullities, eigenvectors corresponding to distinct eigenvalues are linearly independent, How to check whether or not a matrix is diagonalizable, Procedure for diagonalizing a matrix, relation between P and D in the diagonalization, Geometric and algebraic multiplicities of a eigenvalue and their characterization of diagonalizability of a matrix.
    Inner product on a vector space, Inner product spaces, 3 different Inner products on R^n, Relation between different inner products on R^n. Inner products on Matrices, Polynomials, and Continuous functions, Norm and distance functions and their properties, Geometry from an i.p.s., Cauchy-Schwarz inequality, Triangle inequality, Angle between two vectors in an i.p.s., Orthogonal vectors, Generalized Pythagoras Theorem, Orthogonal complement of a subspace, Properties and examples of Orthogonal complements, Null(A) and Row(A) are orthogonal complements, Finding the basis of an orthogonal complement in the Euclidean space, Orthogonal and Orthonormal sets of vectors, Orthonormal Basis, Coordinate vector relative to an Orthonormal basis, Projection theorem: orthogonal projection onto a subspace, Gram-Schmidt process for creating an Orthonormal basis of an inner product space with an example. (From Sections 5.2, 6.1, 6.2 and 6.3)
    
    An overview of: Characterization of linearity in functions, Linear transformations from R^n to R^m and its relation to matrix multiplication with an mxn matrix, zero transformation, Identity operator, Reflection operator as a linear operator, More examples of linear operators, Compositions of linear transforms, Injective and surjective(onto) linear transforms, Characterization of invertible matrices in terms of their corresponding linear transforms, Inverse of a linear transform - when does it exist and how to find it.
    Discussion of EXAM #3 solutions during Special Zoom Office Hour with video recording.
    
    Missed on following topics due to 3 extra holidays on Mondays and Wednesdays this semester: QR decomposition of a matrix; Best approximation in an R^3 and in any ips, Best Approximation Theorem, Least squares problem, Derivation and Consistency of Normal system and usage of least square solutions to find the projection of a vector onto a subspace; Orthogonal Matrices and Orthogonal Diagonalization.
  
  
  • Homework: All HW problems below are based on the 12th edition of the textbook. If you are using an earlier edition, please make sure you are solving the correct problems.
    It is essential that you complete the Reading HWs to keep up with the class.
    You only have to submit solutions to `Submission Problems'. However, solving a majority of the `Suggested Problems' is strongly encouraged. Solving these problems will improve your understanding of the course material and better prepare you for the exams.
    Work on the HW problems before and over the weekend so that you ask for help on Canvas forums or in-person with the instructor or the TA during office hours before Wednesday.
    • Reading HW (not for submission):
      Read Examples 4, 5, 7, and 12 in Section 6.1.
      Read Examples 2, 3, and 6 in Section 6.2.
      Read Examples 1, 2, and 3 in Section 6.3.
      Read Example 8 in Section 6.3.
    • Suggested Problems for additional practice (not for submission): Section 6.1: #2, #5, #10, #33, #35, #TF. Section 6.2: #1, #3, #5, #11, #45. Section 6.3: #7, #31, #38.
    • HW#11 for Submission: Due 11:59pm, Wednesday, 11/27. Submit a PDF file through Canvas Assignment. HW solutions to be distributed.
      You are required to follow the "HW Discussion and Solution Rules" from the Course Information Handout.
      Solve each of the following problems.
      Section 6.1: two of (#18, #20, #22), #28, #34.
      Section 6.2: two of (#17, #25, #27, #37), one of (#41, #46, #47).
      Section 6.3: one of (#3b, #4a), one of (#5, #10), two of (#29, #36, #37).

• Many interesting articles on Linear Algebra and its cutting edge development at Quanta Magazine.
• Use/Purpose of Linear Algebra, by Oliver Knill (Harvard).
• Student companion site for the textbook
• A Self-Guided Aid to Proofs, by Daniel Solow: Old version in HTML; New version in DOC.
• Linear Algebra Toolkit for practice and play.